
 there is a trace morphism \(\textup {Tr}_f \colon R^{2d} f_! \mathbb {Q}_X(d) \rightarrow \mathbb {Q}_T\)  which is an isomorphism if and only if all of the fibers of \(f\)  have a unique irreducible component of dimension \(d\)  (cf. [1]}),

 if \(f\)  has reduced fibers, then the sheaf \(R^{2d}f_!\mathbb {Q}_X(d)\)  is the \(\mathbb {Q}\) -linearization of the sheaf of sets of irreducible components of the fibers of \(f\)  (cf. [2]}).


